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A213551 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

%I A213551 #9 Jul 15 2012 04:09:54
%S A213551 1,6,3,21,15,6,56,46,28,10,126,111,81,45,15,252,231,186,126,66,21,462,
%T A213551 434,371,281,181,91,28,792,756,672,546,396,246,120,36,1287,1242,1134,
%U A213551 966,756,531,321,153,45,2002,1947,1812,1596,1316,1001,686,406
%N A213551 Rectangular array:  (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.
%C A213551 Principal diagonal:  A213552
%C A213551 Antidiagonal sums:  A051923
%C A213551 Row 1,  (1,3,6,...)**(1,3,6,...): A000389
%C A213551 Row 2,  (1,3,6,...)**(3,6,10,...): (k^5 + 15*k^4 + 85*k^3 + 165*k^2 + 94*k)/120
%C A213551 Row 3,  (1,3,6,...)**(6,10,15,...): (k^5 + 20*k^4 + 155*k^3 + 340*k^2 + 204*k)/120
%C A213551 For a guide to related arrays, see A213500.
%H A213551 Clark Kimberling, <a href="/A213551/b213551.txt">Antidiagonals n = 1..60, flattened</a>
%F A213551 T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).
%F A213551 G.f. for row n:  f(x)/g(x), where f(x) = n*(n+1) - 2*((n-1)^2)*x + 2*(n-1)*x^2 and g(x) = 2*(1 - x)^2.
%e A213551 Northwest corner (the array is read by falling antidiagonals):
%e A213551 1....6....21....56....126....252
%e A213551 3....15...46....111...231....434
%e A213551 6....28...81....186...371....672
%e A213551 10...45...126...281...546....966
%e A213551 15...66...181...396...756....1316
%e A213551 21...91...246...531...1001...1722
%t A213551 b[n_] := n (n + 1)/2; c[n_] := n (n + 1)/2
%t A213551 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213551 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213551 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213551 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213551 *)
%t A213551 d = Table[t[n, n], {n, 1, 40}] (* A213552 *)
%t A213551 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213551 s1 = Table[s[n], {n, 1, 50}] (* A051923 *)
%Y A213551 Cf. A213500.
%K A213551 nonn,tabl,easy
%O A213551 1,2
%A A213551 _Clark Kimberling_, Jun 17 2012